PQStat - Baza Wiedzy

LOWESS (locally weighted scatterplot smoothing) also known as LOESS (locally estimated scatterplot smoothing) is one of many „modern” modeling methods based on the least squares method. LOESS combines the simplicity of linear regression with the flexibility of nonlinear regression. Locally weighted regression (LOESS) was independently introduced in several different fields in the late 19th and early 20th centuries (Henderson, 1916¹⁾; Schiaparelli, 1866²⁾).In the statistical literature, the method was introduced independently from different perspectives in the late 1970s by Cleveland, 1979³⁾, among others. The method used in the PQstat program is based on this particular work. The basic principle is that a smooth function can be well approximated by a low-degree polynomial (the program uses a linear function i.e., a first-degree polynomial) in the neighborhood of any point x.

Procedure algorithm:

(1)

For each point in the dataset, we build a window containing adjacent elements. The number of elements in the window is determined by the smoothing parameter . The higher its value, the smoother the function will be. If this parameter is e.g. 0.2, then about 20\% of the data will be in the window and they will be used to build the polynomial (here the unicomial, i.e. the linear function).

Due to the need to maintain the symmetry of the window, i.e., the data in the window should be above and below the point for which we are building the model, the number of elements in the window should be odd. Therefore, the number of elements obtained from the parameter is rounded up to the first odd number. This produces a window containing an element and the corresponding number of elements before and after that element in an ordered collection of data.

For example, if the window is to contain 7 elements, it will contain the element and the three preceding and three following elements of the sample. The window determined for the first and last sample elements is of the same size but the test element is not symmetrically placed in it i.e. in the middle.

(2)

At each point in the dataset, a low-degree polynomial (here a linear function) is fitted to a subset of the data located in the window. The fit is performed using a weighted least squares method, which gives more weight to points near the point whose response is estimated and less weight to points further away. In this way, a different polynomial function formula (here a linear function) is assigned to each point. The weights used in the weighted least squares method can be set quite flexibly, but points distant from the set must have less weight than points nearby. Here the weights proposed by Cleveland, the so-called tricube, are used

where the distance from point was 0 -for points outside the designated window and was given as the actual distance between points, but normalized to the interval [0,1]-for points located within the window, so that the maximum distances in all windows were the same.

(3)

At each point, the value of the function is computed based on the polynomial formula (here a linear function) assigned to it. Based on the points and the points estimated by this method, a smoothed function is drawn to fit the data.

The estimation of the regression function by kernel smoothing is sometimes called the Nadaraya-Watson estimation (Nadaraya, 1964⁴⁾; Watson, 1964⁵⁾). The kernel estimate is a weighted average of the observations within the smoothing window:

where is the kernel function described in section Kernel estimation

The smoothing parameter (bandwidth) has a decisive influence on the obtained estimator. The higher the value of the smoothing parameter, the greater the degree of smoothing. It is possible to choose any smoothing parameter by setting a user value. It is also possible to select it automatically by SNR, SROT or OS method. Kernel function has much less influence on the obtained result. We can choose kernels: Gaussian, uniform function (rectangle), triangular, Epanechnikov, quartic or biweight (fourth degree). A description of the different magnitudes of the smoothing parameter and kernel function can be found in the aforementioned chapter.

The window with the settings of the point plot options with the fitted function by the LOWESS method or by kernel smoothing can be found in various analyses. You can also make this graph via the menu Plots→Scatter Plot.